The main difficulty of this scheme is how to estimate \(\bf {G}\) , since all RIS elements are passive without signal processing capability. To achieve this goal, [1]} proposed a dual-link pilot transmission strategy as mentioned before. The key idea is that, the BS firstly transmits pilot signals to the RIS via the downlink channel \({\bf {G}}^T\) , and then the RIS reflects pilot signals back to the BS via the uplink channel \(\bf {G}\) . There are \((N+1)\)  sub-frames for the dual-link pilot transmission, where each sub-frame consists of \(M\)  time slots. In the \(m_1\) th time slot (\(m_1=1,2,\cdots ,M\) ) of the \(t\) th sub-frame (\(t=1,2,\cdots ,N+1\) ), the \(m_1\) th antenna of the BS transmits the pilot signal \(s_{m_1,t}\)  to the RIS and other \((M - 1)\)  antennas of the BS receive the pilot signal reflected by the RIS. The received pilot signal \(y_{m_1,m_2,t}\)  at the \(m_2\) th antennas of the BS can be represented as \((m_2=1,2,\cdots ,M, m_1\ne m_2)\)
\(\begin{aligned}y_{m_1,m_2,t}=&\left[{\bf {g}}_{m_2}^T{\rm {diag}}{({\mathbf {\theta }}_t)}{\bf {g}}_{m_1}+z_{m_1,m_2}\right]s_{m_1,t}+w_{m_1,m_2,t}\\ = &\left[{\bf {g}}_{m_2}^T{\rm {diag}}{({\bf {g}}_{m_1})}{\mathbf {\theta }}_t+z_{m_1,m_2}\right]s_{m_1,t}+w_{m_1,m_2,t},\end{aligned}\) 
